So from Interlaken to Luzern you will cover a small part of this route - this may be enough for you though.

If you want to do the entire route you should travel from Zermatt to Montreux and then on to Interlaken and Grindelwald. Then you can complete the route when going to Luzern before heading to Paris.

You can use the Swiss railways timetable here to plan out your routes and schedules with the trains.

You don't have a lot of time to do these two train routes and visit the towns you mention. Perhaps Zurich - St. Moritz (1 night) - Zermatt (1 night) - Montreux - Interlaken (1 night) - Luzern (1 night). I'm not sure if that gives you enough time to do anything else you want to do though.